Electron doping and superconductivity in the two-dimensional Hubbard model

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Electron doping and superconductivity in the two-dimensional Hubbard model

D. Eichenberger and D. Baeriswyl

Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland

An elaborate variational wave function is used for studying superconductivity in the 共repulsive兲 two- dimensional Hubbard model, including both nearest- and next-nearest-neighbor hoppings. A marked asymmetry is found between the “localized” hole-doped region and the more itinerant electron-doped region. Super- conductivity withd-wave symmetry turns out to be restricted to densities where the Fermi surface crosses the magnetic zone boundary. A concomitant peak in the magnetic structure factor at 共␲,␲兲 clearly points to a magnetic mechanism.

One of the central issues in the ﬁeld of high-temperature superconductors has been—and for some researchers still is—the question whether pairing in the cuprates arises from purely repulsive interactions, as proposed by Anderson two decades ago.¹This question has been studied extensively in the framework of the two-dimensional共2D兲共repulsive兲Hub- bard model 共and the related t-J model兲. Recent progress, both in dynamical mean-ﬁeld theory^2,3 and in variational calculations,⁴has strengthened the case for the existence of a superconducting phase in the Hubbard model, with a 共d-wave兲gap parameter reasonably close to the experimental values for intermediate interaction strengths 共Uof the order of the bandwidth兲. This conclusion has been challenged on the basis of Monte Carlo simulations,⁵which we believe to be not conclusive, as discussed below.

Most previous studies of the Hubbard model have been restricted to nearest-neighbor hopping, where electron doping does not differ from hole doping. Here we show that the addition of second-neighbor hopping 共which breaks the electron-hole symmetry兲changes this behavior substantially.

While the hole-doped side is “localized” and shows kinetic- energy-driven superconductivity with a large condensation energy, the electron-doped side is itinerant with a potential- energy-driven superconductivity and a small condensation energy.

The Hubbard HamiltonianHˆ=Hˆ

0+UDˆ consists of a hopping term共“kinetic energy”兲

Hˆ

0= −

兺

i,j,␴

tij共ci†␴

cj␴+c^†_j_␴ci␴兲共1兲 and an on-site repulsionUDˆ, whereDˆ=兺ini↑ni↓ is the num- ber of doubly occupied sites, ni␴=c_i^†_␴ci␴, and the operator c_i^†_␴共ci␴兲creates共annihilates兲an electron at siteiwith spin␴^. We use the trial ground state

兩⌿典=e^−hH^ˆ⁰^/^te^−gD^ˆ兩⌿0典, 共2兲 wheregandh are共real兲variational parameters and兩⌿0典is a BCS state with d-wave symmetry. The ﬁrst term in Eq. 共2兲 promotes delocalization and yields a substantial improve- ment with respect to the Gutzwiller wave function共h= 0兲. In fact, it has been demonstrated that ansatz共2兲is very close to the exact ground state both for small 2D systems⁶and for the solvable 1/rchain.⁷

In our previous study of the simple Hubbard model 共tij

=t for nearest-neighbor sites and 0 otherwise兲,⁸ we have found that wave function共2兲exhibitsd-wave pairing below a hole concentration of 0.18. At ﬁrst sight this result seems to contradict recent Monte Carlo simulations,⁵where no signa- ture for superconductivity was found, but a closer look shows that there is no discrepancy. In fact, three of the four hole densities considered in Ref.5共0.18, 0.22, 0.28兲are in a region where our variational ground state is also nonsuper- conducting. For the remaining hole density of 0.09 the order parameter共for U/t= 6, the lattice size is 8⫻8, as chosen in Ref.5兲is likely to be too small⁴to be detected by the simulation.

The restriction to nearest-neighbor hopping leads to a 共bare兲Fermi surface which disagrees qualitatively with photoemission experiments on layered cuprates.⁹ Therefore we consider from now on the more realistic case of both nearest- 共t兲 and next-nearest-neighbor hoppings 共t⬘^兲 with canonical parameters U= 8t and t⬘= −0.3t. The bare single-particle spectrum,

⑀kជ= − 2t共coskx+ cosky兲− 4t⬘^cos^kxcosky,

leads to the Fermi surfaces of Fig. 1. The innermost line corresponds to the Van Hove ﬁlling where the Fermi surface passes through the saddle points at 共␲, 0兲 and共0 ,␲兲. These crossings between the Fermi surface and the magnetic zone boundary共the “hot spots”兲 move inward as the densityn is increased and ﬁnally merge 共outermost line兲. Hot spots are

k_x k_y

n = 1.207 n = 1.000 n = 0.726 (0,0)

(0,π)

(π,0) Hot spots

FIG. 1. 共Color online兲Fermi surface for three different electron densities.

Published in "Physical Review B 79(10): 100510(R), 2009"

which should be cited to refer to this work.

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restricted to 0.726⬍n⬍1.206. Our results, to be discussed below, indicate that superconductivity occurs only in this range.

The BCS state兩⌿0典in Eq.共2兲introduces, in addition tog andh, two other parameters: the amplitude ⌬of the superconducting gap function and the “chemical potential”␮. To compute the variational energy, the exponent of the operator e^−gD^ˆ is first decoupled by applying a discrete Hubbard- Stratonovich transformation,¹⁰ which introduces an Ising spin at each site. All operators are then quadratic in creation and annihilation operators, and therefore the fermionic de- grees of freedom can be integrated out. The remaining sum over Ising spin configurations is performed by a Monte Carlo simulation. In order to avoid the minus sign problem, calculations are carried out in the grand canonical ensemble with an average density fixed by␮, which is therefore not a variational parameter. To reduce the statistical error, the optimiza- tion procedure is based on the method proposed by Ceperley et al.¹¹ and Umrigar et al..¹² We have used periodic- antiperiodic boundary conditions.

The minimization of the energyE for ﬁxed average densities yields the results of Tables IandIIfor hole and electron dopings, respectively. We notice that the parameter g, which controls the crossover between itinerant共smallg兲and localized 共large g兲 many-particle states, remains large for hole doping but decreases rapidly for electron doping. There- fore, while the hole-doped region 0.75⬍n⬍0.95 is a localized doped Mott insulator, the electron-doped part 1.05⬍n

⬍1.2 rapidly undergoes a crossover to an itinerant regime as n increases. 共We estimate a crossover parameter g_cr⬇3.兲 This striking difference may root in the bare single-particle density of states at the Fermi energy, which is much larger for hole than for electron doping. We notice that our wave

function is excellent for the itinerant regime and somewhat less reliable for the localized regime.⁷

Figures 2 and3 show, respectively, the gap parameter⌬ and the superconducting order parameter ⌽=兩具ci†↑

c_j

i↓

† 典兩 as functions of doping concentration x= 1 −n. The correspond- ing results for t⬘^{= 0} ^共Ref. ^8兲 are completely electron-hole symmetric and are reproduced only in the right panels. On the hole-doped side superconductivity exists for 0⬍x

⬍0.25共⌬remains ﬁnite, but⌽vanishes forx→0兲, i.e., in a larger region than for t⬘= 0. In contrast, on the electron- doped side the superconducting region is reduced to −0.2

⬍x⬍−0.05. Thus we ﬁnd indeed that superconductivity is restricted to densities where the共bare兲Fermi surface passes through hot spots共see Fig.1兲. Remarkably, for electron doping the gap is suppressed very close to half ﬁlling even in the absence of competing antiferromagnetic long-range order.

The marked difference between electron and hole dopings is conﬁrmed by the condensation energy E_cond=E共0兲−E共⌬兲, which is 1 order of magnitude smaller for electron doping than for hole doping, as depicted in Fig.4.

TABLE I. Chemical potential, parametersgandh, and energy per site for hole doping共n⬍1兲and an 8⫻8 lattice.

n ␮ g h E/t

0.7500 −0.9921共1兲 4.2共1兲 0.113共2兲 −0.858共1兲 0.7800 −0.9612共3兲 4.0共1兲 0.112共2兲 −0.829共1兲 0.8125 −0.9107共3兲 3.9共1兲 0.111共2兲 −0.795共1兲 0.8400 −0.788共1兲 3.7共1兲 0.111共2兲 −0.763共1兲 0.9000 −0.728共1兲 3.8共1兲 0.111共2兲 −0.676共1兲 0.9500 −0.603共1兲 4.0共1兲 0.114共2兲 −0.591共1兲

TABLE II. Chemical potential, parametersgandh, and energy per site for electron doping共n⬎1兲and an 8⫻8 lattice.

n ␮ g h E/t

1.0500 0.7666共1兲 3.6共1兲 0.109共2兲 −0.222共1兲 1.0800 0.6440共1兲 3.4共1兲 0.106共2兲 −0.069共1兲 1.1000 0.5488共1兲 3.2共1兲 0.104共2兲 0.040共1兲 1.1300 0.4380共1兲 3.0共1兲 0.100共2兲 0.206共1兲 1.1600 0.3870共1兲 2.9共2兲 0.096共3兲 0.374共1兲 1.2000 −0.2996共1兲 2.6共2兲 0.091共3兲 0.608共1兲

0 0.1 0.2 0 0.3

0.02 0.04 0.06 0.08 0.1 0.12 0.14

t’ = -0.3t, 8x8

0 0.1 0.2 0.3

t’ = -0.3t, 10x10 t’ = 0, 8x8

Hole-doped Electron-doped

Δ/t

x = 1 - n

- - -

FIG. 2.共Color online兲Optimized gap parameter as a function of doping for two different lattice sizes. For comparison, the result for t⬘= 0 is also shown共from Ref.8兲.

0 0.1 0.2 0 0.3

0.01 0.02 0.03 0.04 0.05 0.06

0 0.1 0.2 0.3

t’ = -0.3t, 8x8 t’ = 0, 8x8 Φ= < c_i⁺c_i+τ⁺>

x = 1 - n

- - -

FIG. 3. 共Color online兲 Superconducting order parameter as a function of doping for an 8⫻8 lattice.

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Figure5shows the kinetic, potential, and total energies as functions of the gap parameter for the two densities n

= 0.78 and 1.16. For hole doping the energy gain is clearly due to a decrease in kinetic energy, while for electron doping the decrease in potential energy gives the main contribution to the condensation energy. The same behavior has been con- sistently obtained for other densities. The quantity

W= − 2

兺

kជ

⳵²⑀kជ

⳵^kx

2具n_kជ典, 共3兲

also plotted in Fig.5, is—up to a minus sign—proportional to the integrated optical conductivity originating from intraband transitions.¹³For the simple Hubbard model 共t⬘^{= 0}兲 W is equal to the kinetic energy, but fort⬘^⫽0 the two quantities differ. For hole dopingWhas a pronounced minimum in the region of the optimal gap, corresponding to an increase in oscillator strength, while for electron doping W increases monotonically with⌬, akin to the BCS behavior where the

oscillator strength is reduced at the onset of superconductivity.

We attribute this asymmetry between hole and electron dopings to the different values of the correlation parameterg 共see TablesIandII兲, which puts the hole-doped system into the localized regime, while the electron-doped system is more itinerant. To make the point clear we consider the simple Hubbard model共t⬘^{= 0兲}both in the smallU共itinerant兲 and largeU共localized兲limits. In the smallUlimit superconductivity is produced by the Kohn-Luttinger mechanism,¹⁴ where the condensation energy is expected to increase as a function ofU/t, whereas in the largeU limit the condensation energy arises from magnetic exchange and thus is likely to increase with t/U. The change in kinetic energy is then obtained through the Hellman-Feynman theorem,

E_kin共⌬兲−E_kin共0兲= −t⳵

⳵^t^E^cond^, ^共4兲 which is positive in the small U limit and negative in the largeUlimit.

Additional information can be gained from the magnetic structure factor

S共qជ兲= 1 N

兺

i,j

eîq^ជ·共^R^ជⁱ^−R^ជ^j^兲具共n_i_↑−n_i_↓兲共n_j_↑−n_j_↓兲典, 共5兲 displayed in Fig.6.S共qជ兲exhibits a clear maximum at共␲^,␲兲, which is largest close to half filling and decreases as doping increases. There is very little difference between electron and hole dopings, presumably because in the large U limit spin correlations depend mostly through the exchange constants J= 4t²/UandJ⬘^{= 4t}⬘²^/Ûon the microscopic parameters and therefore are essentially independent of the sign oft⬘^.

We have seen above that superconductivity is restricted to the region where two points of the Fermi surface can be connected 共at least approximately兲 by the antiferromagnetic wave vector 共␲^,␲兲. This together with the strong peak of S共qជ兲for qជ=共␲^,␲兲supports the point of view that superconductivity in the two-dimensional Hubbard model is due to a 0

0.1 0.2 0 0.3

0.002 0.004 0.006

0 0.1 0.2 0.3 0

0.01 0.02 0.03

t’ = -0.3t t’ = 0

Electron-doped Hole-doped

- - -

x = 1 - n E_cond/t

FIG. 4. 共Color online兲 Condensation energy per site for an 8⫻8 lattice.

0 0.01 0.02 0.03 Δ/t -1

-0.5 0 0.5 1 1.5

0 0.01 0.02 0.03 Δ/t

-2 -1 0 1 2 3

E_kin/t E_pot/t E/tW/t

n = 0.78

n = 1.16 10^-3

x x10^-3

FIG. 5. 共Color online兲 Changes in kinetic, potential, and total energies as well as in the quantityWas functions of the gap parameter for an 8⫻8 lattice. 共−W is proportional to the oscillator strength for intraband transitions.兲 The relative uncertainties are smaller than the symbol sizes.

0 1 2 3 4 5

S(q)

0.050.10 0.160.20

0 1 2 3 4 5

0.050.10 0.160.22 0.25

(0,0) (π,π)

(π,0) (0,0)

Electron-doped

(0,0) (π,0) (π,π)

Hole-doped

FIG. 6. 共Color online兲Magnetic structure factor as a function of the momentum, for various densities above and below half ﬁlling, for an 8⫻8 lattice.

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magnetic mechanism. This conclusion is now widely ac- cepted, but the question whether the mechanism is of the resonating valence bond 共RVB兲type¹ or arises from the exchange of spin ﬂuctuations¹⁵ is presently under debate.^16,17 We cannot solve this problem on the basis of our variational calculations, but the distinction between localized and itinerant regimes can give additional useful insight. According to our results the former is appropriate for the hole-doped region, where superconductivity is associated with a decrease in kinetic energy; the latter regime is found for electron doping, where superconductivity arises from a more conven- tional gain in potential energy.

The results described above compare surprisingly well with experiments on cuprates, better than our earlier calculations⁸where only nearest-neighbor hopping has been taken into account. This concerns the phase diagram, in par- ticular, for electron-doped materials for which a recent sys- tematic study¹⁸gives a dome shape共forTc兲strikingly similar to our Fig. 2. Photoemission data give values of the superconducting gap ⌬ in the range 10–20 meV for hole-doped compounds 关La_2−xSr_xCuO_4+y 共LSCO兲共Ref. 19兲 or Bi2201 共Ref. 9兲兴 and ⬃5 meV for an electron-doped compound 关NdCeCO共Ref. 20兲兴. Choosingt= 300 meV 共neutron data兲,

our maximum gap parameters are 15 and 7 meV for hole and electron dopings, respectively. An increase in oscillator strength, predicted by our calculations for hole doping, has been reported on the basis of optical spectroscopy for an underdoped sample,²¹ but the situation is less clear on the overdoped side. We are not aware of corresponding measure- ments on electron-doped materials.

In conclusion, our variational search ford-wave superconductivity in the two-dimensional Hubbard model gives a dif- ferentiated picture, namely, a large共moderate兲correlation parameter for hole 共electron兲 doping, a gain in kinetic 共potential兲energy due to pairing, and a large共small兲conden- sation energy. Our wave function is expected to be better suited for describing the electron-doped region, which is less localized than the hole-doped region. A more elaborate study is needed to improve our understanding of the pseudogap phase observed in the cuprates for weak hole doping.

We are grateful for ﬁnancial support from the Swiss Na- tional Science Foundation through the National Center of Competence in Research “Materials with Novel Electronic Properties-MaNEP.” The computations have been partially performed on the Pléiades cluster of the EPFL.

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